Abstract

We consider the pp-Laplacian operator on a domain equipped with a Finsler metric. After deriving and recalling relevant properties of its first eigenfunction for p>1p>1, we investigate the limit problem as p→1p\to 1.

Author information

Contact details are reproduced from the original publication and may be historical.

Matteo Novaga

Dip. di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

novaga@dm.unipi.it

Keywords

  • p-Laplace
  • eigenfunction
  • Finsler metric
  • Cheeger set
  • anisotropic isoperimetric inequality

Mathematics Subject Classification

35J20, 35J70, 49A52, 49G05, 52A05, 52A40

References

  1. B. Andrews: Volume–preserving anisotropic mean curvature flow, Indiana Univ. Math. J. 50 (2001) 783–827.https://doi.org/10.1512/iumj.2001.50.1853Google Scholar
  2. A. Avinyo: Isoperimetric constants and some lower bounds for the eigenvalues of the p–Laplacian, Nonlinear Anal., Theory Methods Appl. 30 (1997) 177–180.https://doi.org/10.1016/S0362-546X(96)00229-5Google Scholar
  3. G. Bellettini, M. Novaga, M. Paolini: Characterization of facet–breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound. 3 (2001) 415–446.https://doi.org/10.4171/ifb/47Google Scholar
  4. M. Belloni, B. Kawohl: A direct uniqueness proof for equations involving the p-Laplace operator, Manuscr. Math. 109 (2002) 229–231.https://doi.org/10.1007/s00229-002-0305-9Google Scholar
  5. M. Belloni, B. Kawohl: The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞, ESAIM, Control Optim. Calc. Var. 10 (2004) 28–52.https://doi.org/10.1051/cocv:2003035Google Scholar
  6. M. Belloni, V. Ferone, B. Kawohl: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003) 771–783.https://doi.org/10.1007/s00033-003-3209-yGoogle Scholar
  7. M. Belloni, B. Kawohl, P. Juutinen: The p-Laplace eigenvalue problem as p → ∞ in a Finsler metric, J. Eur. Math. Soc. (JEMS) 8 (2006) 123–138.https://doi.org/10.4171/jems/40Google Scholar
  8. H. Brezis: Operateurs Maximaux Monotones, North-Holland, Amsterdam (1973).Google Scholar
  9. V. Caselles, A. Chambolle, M. Novaga: Uniqueness of the Cheeger set of a convex body, Pacific J. Math. 232 (2007) 77–90.https://doi.org/10.2140/pjm.2007.232.77Google Scholar
  10. V. Caselles, A. Chambolle, S. Moll, M. Novaga: A characterization of convex calibrable sets in RN with respect to anisotropic norms, Preprint, Università di Pisa, Pisa (2005); Ann. Inst. H. Poincaré, Anal. Non Linéaire, to appear.Google Scholar
  11. J. Cheeger: A lower bound for the smalles eigenvalue of the Laplacian, in: Problems in Analysis. A Symposium in Honor of Salomon Bochner, R. C. Gunning (ed.), Princeton University Press, Princeton (1970) 195–199.https://doi.org/10.1515/9781400869312-013Google Scholar
  12. G. Cupini, N. Fusco, R. Petti: Hölder continuity of local minimizers, J. Math. Anal. Appl. 235 (1999) 578–597.https://doi.org/10.1006/jmaa.1999.6410Google Scholar
  13. E. Di Benedetto: C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl. 7 (1983) 827–850.https://doi.org/10.1016/0362-546X(83)90061-5Google Scholar
  14. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  15. L. Esposito, C. Trombetti: Convex symmetrization and Pólya-Szegö inequality, Nonlinear Anal., Theory Methods Appl. 56 (2004) 43–62.https://doi.org/10.1016/j.na.2003.07.010Google Scholar
  16. A. Ferone, R. Volpicelli: Convex rearrangement: equality cases in the Pólya-Szegö inequality, Calc. Var. Partial Differ. Equ. 21 (2004) 259–272.https://doi.org/10.1007/s00526-003-0256-3Google Scholar
  17. E. Giusti: Metodi Diretti nel Calcolo delle Variazioni, Unione Matematica Italiana, Bologna (1994).Google Scholar
  18. E. Hebey, N. Saintier: Stability and perturbations of the domain for the first eigenvalue of the 1-Laplacian, Arch. Math. 86 (2006) 340–351.https://doi.org/10.1007/s00013-005-1531-7Google Scholar
  19. P. Hild, I. R. Ionescu, T. Lachand-Robert, I. Rosca: The blocking of an inhomogeneous Bingham fluid. Applications to landslides, M2AN, Math. Model. Numer. Anal. 36 (2002) 1013–1026.https://doi.org/10.1051/m2an:2003003Google Scholar
  20. I. R. Ionescu, T. Lachand-Robert: Generalized Cheeger sets related to landslides, Calc. Var. Partial Differ. Equ. 23 (2005) 227–249.https://doi.org/10.1007/s00526-004-0300-yGoogle Scholar
  21. B. Kawohl, V. Fridman: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Commentat. Math. Univ. Carol. 44 (2003) 659–667.Google Scholar
  22. B. Kawohl, T. Lachand-Robert: Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math. 225 (2006) 103–118.https://doi.org/10.2140/pjm.2006.225.103Google Scholar
  23. B. Kawohl, F. Schuricht: Dirichlet problems for the 1-laplace operator, including the eigenvalue problem, Commun. Contemp. Math. 9 (2007) 525–543.https://doi.org/10.1142/S0219199707002514Google Scholar
  24. J. B. Keller: Plate failure under pressure, SIAM Rev. 22 (1980) 227–228.https://doi.org/10.1137/1022032Google Scholar
  25. N. J. Korevaar: Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983) 603–614.https://doi.org/10.1512/iumj.1983.32.32042Google Scholar
  26. T. Lachand-Robert, É. Oudet: Minimizing within convex bodies using a convex hull method, SIAM J. Optim. 16 (2006) 368–379.https://doi.org/10.1137/040608039Google Scholar
  27. L. Lefton, D. Wei: Numerical approximation of the first eigenpair of thep-Laplacian using finite elements and the penalty method, Numer. Funct. Anal. Optimization 18 (1997) 389–399.https://doi.org/10.1080/01630569708816767Google Scholar
  28. G. M. Lieberman: Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl. 12 (1988) 1203–1219.https://doi.org/10.1016/0362-546X(88)90053-3Google Scholar
  29. A. M. Matei: First eigenvalue for the p-Laplace operator, Nonlinear Anal., Theory Methods Appl. 39 (2000) 1051–1061.https://doi.org/10.1016/S0362-546X(98)00266-1Google Scholar
  30. M. Novaga, E. Paolini: Regularity results for boundaries in R2 with prescribed anisotropic curvature, Ann. Mat. Pura Appl., IV. Ser. 184 (2005) 239–261.https://doi.org/10.1007/s10231-004-0112-xGoogle Scholar
  31. S. Sakaguchi: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14 (1987) 403–421.Google Scholar
  32. R. Schneider: Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993).https://doi.org/10.1017/CBO9780511526282Google Scholar
  33. F. Schuricht: An alternative derivation of the eigenvalue equation for the 1-Laplace operator, Arch. Math. 87 (2006) 572–577.https://doi.org/10.1007/s00013-006-1827-2Google Scholar
  34. D. Singmaster, D. Souppouris: A constrained isoperimetric problem, Math. Proc. Camb. Philos. Soc. 83 (1978) 73–82.https://doi.org/10.1017/S030500410005430XGoogle Scholar
  35. G. Strang: Maximum area with Minkowski measures of perimeter, Proc. Royal Soc. Edinburgh 138A (2008) 189–199.https://doi.org/10.1017/S0308210506000862Google Scholar
  36. H. Takeuchi: On the first eigenvalue of the p–Laplacian in a Riemannian manifold, Tokyo J. Math. 21 (1998) 135–140.https://doi.org/10.3836/tjm/1270041991Google Scholar
  37. P. Tolksdorf: On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Commun. Partial Differ. Equations 8 (1983) 773–817.https://doi.org/10.1080/03605308308820285Google Scholar
  38. P. Tolksdorf: Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equations 51 (1984) 126–150.https://doi.org/10.1016/0022-0396(84)90105-0Google Scholar
  39. E. Zeidler: Nonlinear Functional Analysis and Applications. III: Variational Methods and Optimization, Springer, New York (1984).Google Scholar

Suggested citation

B. Kawohl, M. Novaga. “The p-Laplace Eigenvalue Problem as p Approaches 1 and Cheeger Sets in a Finsler Metric.” Journal of Convex Analysis 15 (2008), No. 3, 623–634. https://doi.org/10.68381/jca15043

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.